Regularity estimates for convex functions in Carnot-Carath\'eodory spaces

TL;DR

This paper establishes regularity estimates for convex functions within Carnot-Carathéodory spaces generated by Hörmander vector fields, utilizing the structure of metric balls and subharmonic function estimates.

Contribution

It provides new regularity estimates for convex functions in Carnot-Carathéodory spaces, advancing understanding of their behavior under Hörmander vector fields.

Findings

Regularity estimates for convex functions in Carnot-Carathéodory spaces.

Use of metric ball structure induced by Hörmander vector fields.

Application of local upper estimates for subharmonic functions.

Abstract

We prove some regularity estimates for a class of convex functions in Carnot-Carath\'eodory spaces, generated by H\"ormander vector fields. Our approach relies on both the structure of metric balls induced by H\"ormander vector fields and local upper estimates for the corresponding subharmonic functions.